Sec 3 G2 Maths Indices, compound interest and standard form worksheet with answers

A free Sec 3 G2 Maths worksheet on indices, compound interest and standard form: 10 questions, 30 marks, short questions first and then structured ones, in the school exam style. The answers and full worked solutions are on the last page.

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Sec 3 G2 Maths worksheet: Indices, compound interest and standard form

Total: 30 marksSuggested time: 50 minutes

Name: ______________________________Class: ____________ Date: ____________

Answer all questions. Show all your working. Give non-exact answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless the question says otherwise.

Section A

(a)Express

120\,000

in standard form.

Answer (a)[1]

(b)Write

7.02 \times 10^{-4}

as an ordinary number.

Answer (b)[1]

Simplify

(3a^{2})^{4}

Answer[2]

The value of a car bought for $40 000 depreciates at 12% per annum. Find its value after 5 years, correct to the nearest cent.

Answer$[2]

(a)Evaluate

4^{\frac{1}{2}}

Answer (a)[1]

(b)Evaluate

256^{-\frac{3}{4}}

Answer (b)[2]

Section B

(a)Express

7\,020\,000

in standard form.

Answer (a)[1]

(b)Calculate

(4.6 \times 10^{6}) + (2.9 \times 10^{5})

, giving your answer in standard form.

Answer (b)[2]

(a)Given that

4 = 2^k

, write down the value of

k

Answer (a)[1]

(b)Hence solve the equation

4^{x + 2} = \dfrac{1}{4}

Answer (b)[2]

Wei Ling invests $10 000 in a bank that pays 5% per annum compound interest, compounded annually.

(a)Find the total amount in the account after 3 years, correct to the nearest cent.

Answer (a)$[2]

(b)A different bank pays 5% per annum simple interest. Find how much more interest Wei Ling earns with compound interest than with simple interest over the 3 years, correct to the nearest cent.

Answer (b)$[2]

(a)Simplify

\dfrac{(2a^{2})^{3} \times 3a^{5}}{4a^{2}}

Answer (a)[2]

(b)Find the value of

8^{\frac{2}{3}}

Answer (b)[2]

Light travels at

3 \times 10^{8}

m/s. The distance from the Sun to the Earth is

1.5 \times 10^{11}

(a)Find the time, in seconds, that light takes to travel from the Sun to the Earth. Give your answer in standard form.

Answer (a)s[2]

(b)Convert your answer to minutes, correct to 3 significant figures.

Answer (b)min[1]

A new car costs $96 000. Its value depreciates by 15% each year.

(a)Find the value of the car after 3 years.

Answer (a)$[2]

(b)Find the percentage of the original value that the car has lost after 3 years, correct to 1 decimal place.

Answer (b)%[2]

END OF PAPER

Answers and worked solutions

Question 1

(a)Answer:

120\,000 = 1.2 \times 10^{5}

[1]

Number of places the point moves:

5

120\,000 = 1.2 \times 10^{5}

(b)Answer:

7.02 \times 10^{-4} = 0.000\,702

[1]

A negative power of 10 makes the number smaller: the point moves left.Number of places the point moves:

4

7.02 \times 10^{-4} = 0.000\,702

Question 2

Answer:

81a^{8}

[2]

Numbers:

3^{4} = 81

a

: multiply the indices,

2 \times 4 = 8

81a^{8}

Question 3

Answer: $21109.28[2]

A = P\left(1 - \dfrac{r}{100}\right)^n

A = 40000\left(1 - \dfrac{12}{100}\right)^{5}

A = 21109.2767\ldots

$21109.28

Question 4

(a)Answer:

2

[1]

\sqrt{4} = 2

2

(b)Answer:

\dfrac{1}{64}

[2]

256^{-\frac{3}{4}} = \left(\dfrac{1}{256}\right)^{\frac{3}{4}}

\sqrt[4]{\dfrac{1}{256}} = \dfrac{1}{4}

\left(\dfrac{1}{4}\right)^{3} = \dfrac{1}{64}

\dfrac{1}{64}

Question 5

(a)Answer:

7.02 \times 10^{6}

[1]

Number of places the point moves:

\blank{6}

7\,020\,000 = 7.02 \times 10^{6}

(b)Answer:

4.89 \times 10^{6}

[2]

2.9 \times 10^{5} = 0.29 \times 10^{6}

(4.6 + 0.29) \times 10^{6}

= 4.89 \times 10^{6}

Question 6

(a)Answer:

2^{2}

, so the index is

2

[1]

4 = 2^{2}

(b)Answer:

x = -3

[2]

2^{2(x + 2)} = 2^{-2}

2(x + 2) = -2

2x + 4 = -2

2x = -6

x = -3

Question 7

(a)Answer: $11576.25[2]

A = P\left(1 + \dfrac{r}{100}\right)^n

A = 10000\left(1 + \dfrac{5}{100}\right)^{3}

A = 11576.25

(b)Answer: $76.25[2]

Compound interest

= 11576.2500\ldots - 10000

= 1576.2500\ldots

Simple interest

= \dfrac{10000 \times 5 \times 3}{100}

= 1500

Difference

= 1576.2500\ldots - 1500

= \$76.25

Question 8

(a)Answer:

6a^{9}

[2]

(2a^{2})^{3} = 8a^{6}

8a^{6} \times 3a^{5} = 24a^{11}

Numbers:

\dfrac{24}{4} = 6

a

11 - 2 = 9

6a^{9}

(b)Answer:

4

[2]

\sqrt[3]{8} = 2

2^{2} = 4

4

Question 9

(a)Answer:

5 \times 10^{2}

s[2]

Time

= \dfrac{\text{distance}}{\text{speed}}

= \dfrac{1.5 \times 10^{11}}{3 \times 10^{8}}

= 0.5 \times 10^{3}

\phantom{} = 5 \times 10^{2}

(b)Answer:

8.33

minutes[1]

500 \div 60 = 8.333\ldots

\phantom{500 \div 60} = 8.33

minutes (3 s.f.)

Question 10

(a)Answer: $58 956[2]

96000 \times 0.85^{3}

= 58\,956

(b)Answer:

38.6\%

[2]

Loss

= 96\,000 - 58\,956

= 37\,044

\dfrac{37\,044}{96\,000} \times 100\% = 38.5875\%

\phantom{\dfrac{37\,044}{96\,000} \times 100\%} = 38.6\%

Sec 3 G2 Maths Indices, compound interest and standard form worksheet with answers

In the print window, choose Save as PDF to download it.

Answers and worked solutions

Question 1

(a)Answer:

120\,000 = 1.2 \times 10^{5}

[1]

Number of places the point moves:

5

120\,000 = 1.2 \times 10^{5}

(b)Answer:

7.02 \times 10^{-4} = 0.000\,702

[1]

A negative power of 10 makes the number smaller: the point moves left.Number of places the point moves:

4

7.02 \times 10^{-4} = 0.000\,702

Question 2

Answer:

81a^{8}

[2]

Numbers:

3^{4} = 81

a

: multiply the indices,

2 \times 4 = 8

81a^{8}

Question 3

Answer: $21109.28[2]

A = P\left(1 - \dfrac{r}{100}\right)^n

A = 40000\left(1 - \dfrac{12}{100}\right)^{5}

A = 21109.2767\ldots

$21109.28

Question 4

(a)Answer:

2

[1]

\sqrt{4} = 2

2

(b)Answer:

\dfrac{1}{64}

[2]

256^{-\frac{3}{4}} = \left(\dfrac{1}{256}\right)^{\frac{3}{4}}

\sqrt[4]{\dfrac{1}{256}} = \dfrac{1}{4}

\left(\dfrac{1}{4}\right)^{3} = \dfrac{1}{64}

\dfrac{1}{64}

Question 5

(a)Answer:

7.02 \times 10^{6}

[1]

Number of places the point moves:

\blank{6}

7\,020\,000 = 7.02 \times 10^{6}

(b)Answer:

4.89 \times 10^{6}

[2]

2.9 \times 10^{5} = 0.29 \times 10^{6}

(4.6 + 0.29) \times 10^{6}

= 4.89 \times 10^{6}

Question 6

(a)Answer:

2^{2}

, so the index is

2

[1]

4 = 2^{2}

(b)Answer:

x = -3

[2]

2^{2(x + 2)} = 2^{-2}

2(x + 2) = -2

2x + 4 = -2

2x = -6

x = -3

Question 7

(a)Answer: $11576.25[2]

A = P\left(1 + \dfrac{r}{100}\right)^n

A = 10000\left(1 + \dfrac{5}{100}\right)^{3}

A = 11576.25

(b)Answer: $76.25[2]

Compound interest

= 11576.2500\ldots - 10000

= 1576.2500\ldots

Simple interest

= \dfrac{10000 \times 5 \times 3}{100}

= 1500

Difference

= 1576.2500\ldots - 1500

= \$76.25

Question 8

(a)Answer:

6a^{9}

[2]

(2a^{2})^{3} = 8a^{6}

8a^{6} \times 3a^{5} = 24a^{11}

Numbers:

\dfrac{24}{4} = 6

a

11 - 2 = 9

6a^{9}

(b)Answer:

4

[2]

\sqrt[3]{8} = 2

2^{2} = 4

4

Question 9

(a)Answer:

5 \times 10^{2}

s[2]

Time

= \dfrac{\text{distance}}{\text{speed}}

= \dfrac{1.5 \times 10^{11}}{3 \times 10^{8}}

= 0.5 \times 10^{3}

\phantom{} = 5 \times 10^{2}

(b)Answer:

8.33

minutes[1]

500 \div 60 = 8.333\ldots

\phantom{500 \div 60} = 8.33

minutes (3 s.f.)

Question 10

(a)Answer: $58 956[2]

96000 \times 0.85^{3}

= 58\,956

(b)Answer:

38.6\%

[2]

Loss

= 96\,000 - 58\,956

= 37\,044

\dfrac{37\,044}{96\,000} \times 100\% = 38.5875\%

\phantom{\dfrac{37\,044}{96\,000} \times 100\%} = 38.6\%